Abstract:
"The work of this Magister’s memory is in the framework of the General Theory of
Relativity of Einstein. After a short introduction, followed by a concise but complete
presentation of general relativity in order to dispense the reader to refer to an external
document support, we tackled the rederivation of some interesting metrics of General
Relativity , especially in the field of cosmology . We considered in turn:
The FLRW metric, satisfying the cosmological principle , that is to say corresponding
to a homogeneous isotropic universe, which is reflected by the invariance under
rotations and translations (flat case ), or under rotations and quasi-translations
(spherical and pseudo- spherical cases) .
Shwarzshild metric, static and spherically symmetric , that is to say invariant under
temporal translations and rotation, in absence of cosmological constant .
The Kottler metric or Schwarzshild -de Sitter, also static and spherically symmetric,
but with cosmological constant
The work is done in two steps:
In a first step and for each considered case , we impose the corresponding symmetries.
The method used is that of the Killing vectors , which is an elegant and rigorous
method to implement symmetries . Symmetries result in constraints on the form of
the metric. During this first stage we do not appeal to Einstein's equations , that is to
say we do not use dynamics .
In a second step , having determined in the previous step the most general form of a
metric satisfying some symmetries, we complete determnation of the metric by
tacking into account the Einstein equations.To this end, we have conjectured some
more or less justified forms for the energy-momentum tensor. In this way, we find
the FLRW metric, the Shwarzshild metric inside and outside of the distribution of
matter, and also Kottler metric inside and outside of the distribution of matter."
